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We already know that the mean of a set of numbers is often called
the average, and is found by adding the numbers together and
dividing by the number of numbers in the list. The geometric mean
also tells us about the "middle" of a data set but we find it using
a different method.

Let's define that in the proportion
$\frac{a}{b}=\frac{c}{d}$
, b and c are called means, while a and d are called the extremes.

When the means of a proportion are the same number, that number is
called the geometric mean of the extremes. So in the following
proportion,

$\frac{p}{x}=\frac{x}{q}$

Cross multiplying gives you
${x}^{2}=pq$
. Taking the square root of both sides, we get
$x=\sqrt{pq}$
as the geometric mean of p and q.

In math, the geometric mean, also called the GM, is the average
value that identifies the central tendency of the set of numbers by
finding the product of their values. Simply put, we multiply the
numbers together and take the nth root of the multiplied numbers,
where n equals the total number of data values given (their
cardinality).

As an example, the geometric mean of a given set of numbers 3 and 2,
the geometric mean is equal to
$\sqrt{3}\times 2$
or
$\sqrt{6}$
, which is approximately 2.449.

The geometric mean is not the same as the arithmetic mean, which for
the numbers 3 and 2 would be
$\frac{3+2}{2}=2.5$
.

The important properties of the geometric mean are as follows:

- The GM for the provided data set is always less than the arithmetic mean for the same data set.
- If each object in the data set is substituted by the GM, the product of the subjects remains the same.

Find the geometric mean of 25 and 9.

Since there are two numbers, the geometric mean of the two numbers
is equal to the square root of their product.

$\mathrm{Geometric\; mean}=\sqrt{25\times 9}$

$=\sqrt{225}$

$=15$

The geometric mean of 25 and 9 is 15. (Note that the arithmetic mean
of 25 and 9 is 17, which is more than the geometric mean, satisfying
one of the important properties of geometric mean.)

Find the geometric mean of 4, 10, and 25

Since there are three numbers, the geometric mean will be the cube
root of their product.

$\mathrm{Geometric\; mean}=\sqrt[3]{4\times 10\times 25}$

$=\sqrt[3]{1000}$

$10$

The geometric mean of 4, 10, and 25 is 10.

Find the geometric mean of 5, 10, 25, and 30

Since there are four numbers, the geometric mean will be the 4th
root of their product.

$\mathrm{Geometric\; mean}=\sqrt[4]{5\times 10\times 25\times 30}$

$=\sqrt[4]{37500}$

$=13.915$

The geometric mean of 5, 10, 25, and 30 is approximately 13.915. The
geometric mean is not always going to be a whole number.

The geometric mean has a lot of advantages and is used in a variety
of fields. For example, it is used in stock indexes to calculate the
annual return on the portfolio. It is also used in finance to find
the average growth rates, which are also referred to as the
compounded annual growth rate. In another field entirely, it is used
in biology to study cell division and bacterial growth. The many
uses of the geometric mean show that it's a useful concept worth
knowing well.

If you are having a hard time understanding how to work out the
geometric mean, you could use the help of a private tutor. A
professional tutor can provide expert assistance as you learn, step
by step, how to calculate the geometric mean and use it in certain
applications. To learn more about how tutoring can help you
understand the geometric mean and other related concepts, contact
the Educational Directors at Varsity Tutors today.

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